x+5/3x+1=x+8/x

Solution for x+5/3x+1=x+8/x equation:

x+5/3x+1=x+8/x

We move all terms to the left:

x+5/3x+1-(x+8/x)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

x+5/3x-(+x+8/x)+1=0

We get rid of parentheses

x+5/3x-x-8/x+1=0

We calculate fractions


x-x+5x/3x^2+(-24x)/3x^2+1=0

We add all the numbers together, and all the variables

5x/3x^2+(-24x)/3x^2+1=0

We multiply all the terms by the denominator


5x+(-24x)+1*3x^2=0

Wy multiply elements

3x^2+5x+(-24x)=0

We get rid of parentheses

3x^2+5x-24x=0

We add all the numbers together, and all the variables

3x^2-19x=0

a = 3; b = -19; c = 0;
Δ = b2-4ac
Δ = -192-4·3·0
Δ = 361
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{361}=19$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-19)-19}{2*3}=\frac{0}{6}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-19)+19}{2*3}=\frac{38}{6}
=6+1/3
$